THE CONICS, BOOK I 
139 
PL is called the latus rectum (opOia) or the 'parameter of 
the ordinates (nap’ fju Suravrai at Karayoyevai rer ay perns) in 
each case. In the case of the central conics, the diameter PP' 
is the transverse (rj nXayia) or transverse diameter; while, 
even more commonly, Apollonius speaks of the diameter and 
the corresponding parameter together, calling the latter the 
latus rectum or erect side (opdia nXevpd) and the former 
the transverse side of the figure (eJSos) on, or applied to, the 
diameter. 
Fundamental properties equivalent to Cartesian equations. 
If p is the parameter, and d the corresponding diameter, 
the properties of the curves are the equivalent of the Cartesian 
equations, referred to the diameter and the tangent at its 
extremity as axes (in general oblique), 
y 2 = px (the parabola), 
y 1 = px + x 2 (the hyperbola and ellipse respectively). 
Thus Apollonius expresses the fundamental property of the 
central conics, like that of the parabola, as an equation 
between areas, whereas in Archimedes it appears as a 
proportion 
y 2 : (a 2 + x 2 ) = b 2 : a 2 , 
which, however, is equivalent to the Cartesian equation 
referred to axes with the centre as origin. The latter pro 
perty with reference to the original diameter is separately 
proved in I. 21, to the effect that QV 2 varies as PV.P'V, as 
is really evident from the fact that QV 2 : PV. P'V = PL : PP', 
seeing that PL: PP' is constant for any fixed diameter PP'. 
Apollonius has a separate proposition (I. 14) to prove that 
the opposite branches of a hyperbola have the same diameter 
and equal latera recta corresponding thereto. As he was the 
first to treat the double-branch hyperbola fully, he generally 
discusses the hyperbola (i.e. the single branch) along with 
the ellipse, and the opposites, as he calls the double-branch 
hyperbola, separately. The properties of the single-branch 
hyperbola are, where possible, included in one enunciation 
with those of the ellipse and circle, the enunciation beginning,